00:01
Hi, in this question we are given with the region r, which is bounded by the graph of function fx which is equal to 9x minus x squared and below by the graph of gx as equal to x over the interval from 2 to 4.
00:20
We need to find the center of mass of the region assuming that the region has a constant density row.
00:27
First, if we plot the graph for this function, we get the region like this.
00:34
And we need to go for below this gx as x, that means this one region.
00:42
So for this, we must go from 2 to 4.
00:47
That means we must consider this part only.
00:51
Now for this part, as we can see, here we can find the area that would be equal to integration from 2.
00:58
2 to 4 fx d x and in this case in this area only we have function as x only so our area would be equal to integration from 2 to 4 x times d x that would be equal to x square by 2 limits from 2 to 4 to 4 2 minus 2 square by 2 and solving this we get our area that would be equal to 6 now solving for the center of mass, we know we can find the x coordinate of center of mass as equal to 1 by a times integration from a to b, x times fx, dx, and y bar to be equal to 1 by a times integration from a to b, half of fx square dx.
01:57
Now in our case our function is x only.
02:00
So our x -par would be equal to 1 by 6 times integration from 2 to 4, x times x times dx...